Objective: Set up and solve the commercial fishing economic optimal control problem. Create a program to optimize and display the results. Estimated Time: 30 minutes

The commercial fishing optimal control problem has an integral profit function that includes the cost of operations and the revenue from fish sales. The population of fish x is influenced by how many fish are removed each year that depends on u. The objective is to maximize the revenue from fishing over a 10 year time period. If there is overfishing (high u) then the returns for subsequent years are reduced and the fish population does not recover. This optimal control problem finds the optimal extraction profile to maximize the commercial fishing profit.

The initial condition of the integral J starts at zero and becomes the integral in the time range of 0 to 10. The end value is maximized at the final point in the time horizon of the optimal control problem. A maximization problem is converted to a minimization problem by multiplying the objective by negative one.

$$\max_{u(t)} J\left(t_f\right) = -\min_{u(t)} J\left(t_f\right)$$

Most optimization solvers require an objective function minimization, including GEKKO and APMonitor.